# L9a2

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## Contents (Knotscape image) See the full Thistlethwaite Link Table (up to 11 crossings). Visit L9a2 at Knotilus! L9a2 is $9^2_{31}$ in the Rolfsen table of links.

### Polynomial invariants

 Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...) $\frac{(t(1)-1) (t(2)-1) \left(t(2)^4-t(2)^3+t(2)^2-t(2)+1\right)}{\sqrt{t(1)} t(2)^{5/2}}$ (db) Jones polynomial $-q^{15/2}+3 q^{13/2}-4 q^{11/2}+6 q^{9/2}-7 q^{7/2}+6 q^{5/2}-6 q^{3/2}+3 \sqrt{q}-\frac{3}{\sqrt{q}}+\frac{1}{q^{3/2}}$ (db) Signature 3 (db) HOMFLY-PT polynomial $z^7 a^{-3} -z^5 a^{-1} +5 z^5 a^{-3} -z^5 a^{-5} -3 z^3 a^{-1} +7 z^3 a^{-3} -3 z^3 a^{-5} +z a^{-3} -z a^{-5} +2 a^{-1} z^{-1} -3 a^{-3} z^{-1} + a^{-5} z^{-1}$ (db) Kauffman polynomial $z^3 a^{-9} +3 z^4 a^{-8} -2 z^2 a^{-8} +4 z^5 a^{-7} -3 z^3 a^{-7} +4 z^6 a^{-6} -3 z^4 a^{-6} -2 z^2 a^{-6} + a^{-6} +4 z^7 a^{-5} -8 z^5 a^{-5} +6 z^3 a^{-5} -2 z a^{-5} - a^{-5} z^{-1} +2 z^8 a^{-4} -2 z^6 a^{-4} -3 z^4 a^{-4} +3 a^{-4} +7 z^7 a^{-3} -24 z^5 a^{-3} +22 z^3 a^{-3} -3 z a^{-3} -3 a^{-3} z^{-1} +2 z^8 a^{-2} -5 z^6 a^{-2} +z^2 a^{-2} +3 a^{-2} +3 z^7 a^{-1} -12 z^5 a^{-1} +12 z^3 a^{-1} -z a^{-1} -2 a^{-1} z^{-1} +z^6-3 z^4+z^2$ (db)

### Khovanov Homology

The coefficients of the monomials $t^rq^j$ are shown, along with their alternating sums $\chi$ (fixed $j$, alternation over $r$).
 \ r \ j \
-3-2-10123456χ
16         11
14        2 -2
12       21 1
10      42  -2
8     32   1
6    34    1
4   33     0
2  25      3
0 11       0
-2 2        2
-41         -1
Integral Khovanov Homology $\dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}$ $i=2$ $i=4$ $r=-3$ ${\mathbb Z}$ $r=-2$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2$ ${\mathbb Z}$ $r=-1$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=0$ ${\mathbb Z}^{5}\oplus{\mathbb Z}_2$ ${\mathbb Z}^{3}$ $r=1$ ${\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=2$ ${\mathbb Z}^{4}\oplus{\mathbb Z}_2^{3}$ ${\mathbb Z}^{3}$ $r=3$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{4}$ ${\mathbb Z}^{4}$ $r=4$ ${\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=5$ ${\mathbb Z}\oplus{\mathbb Z}_2^{2}$ ${\mathbb Z}^{2}$ $r=6$ ${\mathbb Z}_2$ ${\mathbb Z}$

### Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory. See A Sample KnotTheory Session.

### Modifying This Page

 Read me first: Modifying Knot Pages See/edit the Link Page master template (intermediate). See/edit the Link_Splice_Base (expert). Back to the top.